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A lower bound for graph energy in terms of minimum and maximum degrees
, Article Match ; Volume 86, Issue 3 , 2021 , Pages 549-558 ; 03406253 (ISSN) ; Ghahremani, M ; Hosseinzadeh, M. A ; Ghezelahmad, S. K ; Rasouli, H ; Tehranian, A ; Sharif University of Technology
University of Kragujevac, Faculty of Science
2021
Abstract
The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631{633) it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree Δ (G) whose adjacency matrix is non-singular, E(G) +δ (G) + Δ (G) and the equality holds if and only if G is a complete graph. Here, we prove the validity of this conjecture for planar graphs, triangle-free graphs and quadrangle-free graphs. © 2021 University of Kragujevac, Faculty of Science. All rights reserved
Upper bounds on the energy of graphs in terms of matching number
, Article Applicable Analysis and Discrete Mathematics ; Volume 15, Issue 2 , 2021 , Pages 444-459 ; 14528630 (ISSN) ; Alazemi, A ; Andelic, M ; Sharif University of Technology
University of Belgrade
2021
Abstract
The energy of a graph G, ϵ(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number µ(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ∆ ≥ 6 we present two new upper bounds for the energy of a graph: (Formula presented) and (Formula presented). The latter one improves recently obtained bound (Formula presented) where ∆e stands for the largest edge degree and a = 2(∆e + 1). We also present a short proof of this result and several open problems. © 2021
Upper bounds on the energy of graphs in terms of matching number
, Article Applicable Analysis and Discrete Mathematics ; Volume 15, Issue 2 , 2021 , Pages 444-459 ; 14528630 (ISSN) ; Alazemi, A ; Andelic, M ; Sharif University of Technology
University of Belgrade
2021
Abstract
The energy of a graph G, ϵ(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number µ(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ∆ ≥ 6 we present two new upper bounds for the energy of a graph: (Formula presented) and (Formula presented). The latter one improves recently obtained bound (Formula presented) where ∆e stands for the largest edge degree and a = 2(∆e + 1). We also present a short proof of this result and several open problems. © 2021
Energy of Graphs
, M.Sc. Thesis Sharif University of Technology ; Akbari Feizabadi, Saeed (Supervisor) ; Fahmideh Gholami, Mahdi (Co-Supervisor)
Abstract
Energy of graphs first defined by Ivan Gutman in 1978[1]. Let G be a graph with (0,1)-adjacency matrix A and let λ_1≥⋯≥λ_n be eigenvalues of A. Grap h energy is defined as the sum of absolute values of the eigenvalues of A and is shown by ɛ(G). let H_1,…,H_k be the vertex-disjoint induced subgraphs of graph G, it is proved that energy of G is at least equal to the sum of energy of H_i subgraphs, where the summation is over i. Also by partitioning edges of G to L_1,…,L_k subgraphs, energy of G is at most equal to sum of energy of L_i subgraphs , where the summation is over i. In this thesi s we study energy of graphs, specially...
Edge addition, singular values, and energy of graphs and matrices
, Article Linear Algebra and Its Applications ; Volume 430, Issue 8-9 , 2009 , Pages 2192-2199 ; 00243795 (ISSN) ; Ghorbani, E ; Oboudi, M. R ; Sharif University of Technology
2009
Abstract
The energy of a graph/matrix is the sum of the absolute values of its eigenvalues. We investigate the result of duplicating/removing an edge to the energy of a graph. We also deal with the problem that which graphs G have the property that if the edges of G are covered by some subgraphs, then the energy of G does not exceed the sum of the subgraphs' energies. The problems are addressed in the general setting of energy of matrices which leads us to consider the singular values too. Among the other results it is shown that the energy of a complete multipartite graph increases if a new edge added or an old edge is deleted. © 2008 Elsevier Inc. All rights reserved
Some lower bounds for the energy of graphs
, Article Linear Algebra and Its Applications ; Volume 591 , 2020 , Pages 205-214 ; Ghodrati, A. H ; Hosseinzadeh, M. A ; Sharif University of Technology
Elsevier Inc
2020
Abstract
The singular values of a matrix A are defined as the square roots of the eigenvalues of A⁎A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A=(BDD⁎C), then E(A)≥2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H)≤E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known...
Energy of Graphs
, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saeid (Supervisor)
Abstract
Let G be a graph with adjacency matrix A and Δ be a diagonal matrix whose diagonal entries are the degree sequence of G. Then the matrices L = Δ− A and Q = Δ+A are called Laplacian matrix and signless Laplacian matrix of G, respectively. The eigenvalues of A, L, and Q are arranged decreasingly and denoted by λ1 ≥ · · · ≥ λn, μ1 ≥ · · · ≥ μn ≥ 0, and q1 ≥ · · · ≥ qn ≥ 0, respectively. The energy of a graph G is defined as E(G) :=
n
i=1
|λi|.
Furthermore, the incidence energy, the signed incidence energy, and the H¨uckel energy of G are
defined as
IE(G) :=
n
i=1
√
qi, LE(G) :=
n
i=1
√
μi, HE(G) :=
2
r
i=1 λi, n=...
n
i=1
|λi|.
Furthermore, the incidence energy, the signed incidence energy, and the H¨uckel energy of G are
defined as
IE(G) :=
n
i=1
√
qi, LE(G) :=
n
i=1
√
μi, HE(G) :=
2
r
i=1 λi, n=...
Bounds for the Energy of Graphs
, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saieed (Supervisor)
Abstract
The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G) ⩾ n. Here, we improve this result by showing that if G is a connected subcubic graph of order n, n ⩾ 8, then E(G) ⩾ 1.01n. Also, we prove that if G is a traceable subcubic graph of order n,then E(G) ⩾ 1.1n. Let G be a connected cubic graph of order n, it is shown that E(G) > n + 2, for n ⩾ 8 and we introduce an infinite family of connected cubic graphs whose for each element, say G, E(G) ⩾ 1.24n, and some important conjectures will be raised about this. At the end, for a graph G and its vertex induced subgraphs H and K,...
Some Bounds on Randić Index of Graphs
, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract
In 1975 a Chemist Milan Randić proposed a concept named Randić index which is defined as follows: This index is generalized by replacing any real number α with which is called the general Randić index. Let G be a graph of order n. Erdős and Bollobás showed the lower bound for Randić index, Also, an upper bound for Randić index is n/2. In 2018 Suil O and Yongtang Shi proved a lower bound with minimum and maximum degree of a graph. They have shown for graph G we have, R(G) Also, a relation between Randić index and the energy of the graph has found. Indeed, it was proved that E(G) ⩾ 2R(G), where E(G) is the energy of graph. Many important bounds related to graph parameters for Randić index...
Some relations between rank, chromatic number and energy of graphs
, Article Discrete Mathematics ; Volume 309, Issue 3 , 2009 , Pages 601-605 ; 0012365X (ISSN) ; Ghorbani, E ; Zare, S ; Sharif University of Technology
2009
Abstract
The energy of a graph G, denoted by E (G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and rank (G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E (G) = rank (G). Among other results we show that apart from a few families of graphs, E (G) ≥ 2 max (χ (G), n - χ (over(G, -))), where n is the number of vertices of G, over(G, -) and χ (G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E (G) in terms of rank (G) are given. © 2008 Elsevier B.V. All rights reserved
Choice number and energy of graphs
, Article Linear Algebra and Its Applications ; Volume 429, Issue 11-12 , 2008 , Pages 2687-2690 ; 00243795 (ISSN) ; Ghorbani, E ; Sharif University of Technology
2008
Abstract
The energy of a graph G, denoted by E (G), is defined as the sum of the absolute values of all eigenvalues of the adjacency matrix of G. It is proved that E (G) ≥ 2 (n - χ (over(G, -))) ≥ 2 (ch (G) - 1) for every graph G of order n, and that E (G) ≥ 2 ch (G) for all graphs G except for those in a few specified families, where over(G, -), χ (G), and ch (G) are the complement, the chromatic number, and the choice number of G, respectively. © 2007 Elsevier Inc. All rights reserved
On the energy of line graphs
, Article Linear Algebra and Its Applications ; Volume 636 , 2022 , Pages 143-153 ; 00243795 (ISSN) ; Alazemi, A ; Anđelić, M ; Hosseinzadeh, M. A ; Sharif University of Technology
Elsevier Inc
2022
Abstract
The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree δ(G) whose adjacency matrix is non-singular, E(G)≥Δ(G)+δ(G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E(G). In this paper, first we show that E(L(G))≥|E(G)|+Δ(G)−5, where L(G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. © 2021 Elsevier Inc
A graph weighting method for reducing consensus time in random geographical networks
, Article 24th IEEE International Conference on Advanced Information Networking and Applications Workshops, WAINA 2010, 20 April 2010 through 23 April 2010, Perth ; 2010 , Pages 317-322 ; 9780769540191 (ISBN) ; Sharif University of Technology
2010
Abstract
Sensor networks are increasingly employed in many applications ranging from environmental to military cases. The network topology used in many sensor network applications has a kind of geographical structure. A graph weighting method for reducing consensus time in random geographical networks is proposed in this paper. We consider a method based on the mutually coupled oscillators for providing general consensus in the network. In this way, one can relate the consensus time to the properties of the Laplacian matrix of the connection graph, i.e. to the second smallest eigenvalue (algebraic connectivity). Our weighting algorithm is based on the node and edge between centrality measures. The...